From a stochastic maximal inequality to infinite-dimensional martingales

As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a stochastic maximal inequality derived by using the Taylor expansion, is presented. The inequality dealing with finite-dimensional discrete-time martingales is pulled up to infinite-dimensional ones by using the monotone convergence arguments. The main results are some weak convergence theorems for sequences of separable random fields of discrete-time martingales under the uniform topology with the help also of entropy methods. As special cases, some new results for i.i.d. random sequences, including a new Donsker theorem and a moment bound for suprema of empirical processes indexed by classes of sets or functions, are obtained.